Dynamical response of hyper-elastic cylindrical shells under periodic load

  • Jiu-sheng Ren (任九生)Email author


Dynamical responses, such as motion and destruction of hyper-elastic cylindrical shells subject to periodic or suddenly applied constant load on the inner surface, are studied within a framework of finite elasto-dynamics. By numerical computation and dynamic qualitative analysis of the nonlinear differential equation, it is shown that there exists a certain critical value for the internal load describing motion of the inner surface of the shell. Motion of the shell is nonlinear periodic or quasi-periodic oscillation when the average load of the periodic load or the constant load is less than its critical value. However, the shell will be destroyed when the load exceeds the critical value. Solution to the static equilibrium problem is a fixed point for the dynamical response of the corresponding system under a suddenly applied constant load. The property of fixed point is related to the property of the dynamical solution and motion of the shell. The effects of thickness and load parameters on the critical value and oscillation of the shell are discussed.

Key words

hyper-elastic cylindrical shells nonlinear differential equation periodic oscillation quasi-periodic oscillation critical load 

Chinese Library Classification


2000 Mathematics Subject Classification



  1. [1]
    Fu Y B, Ogden R W. Nonlinear elasticity[M]. Cambridge: Cambridge University Press, 2001.Google Scholar
  2. [2]
    Beatty M F. Topics in finite elasticity[J]. Applied Mechanics Review, 1987, 40(12):1699–1734.Google Scholar
  3. [3]
    Gent A N. Elastic instability in rubber[J]. Int J Non-Linear Mech, 2005, 40(2):165–175.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Gent A N. Elastic instability of inflated rubber shells[J]. Rubber Chem Tech, 1999, 72(2):263–268.Google Scholar
  5. [5]
    Needleman A. Inflation of spherical rubber balloons[J]. Int J Solids Struct, 1977, 13(3):409–421.CrossRefGoogle Scholar
  6. [6]
    Haughton D M, Ogden R W. On the incremental equations in non-linear elasticity-II: bifurcation of pressurized spherical shells[J]. J Mech Phys Solids, 1978, 26(1):111–138.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Haughton D M, Ogden R W. Bifurcation of inflated circular cylinders of elastic material under axial loading-II: exact theory for thick-walled tubes[J]. J Mech Phys Solids, 1979, 27(4):489–512.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Ren Jiusheng, Cheng Changjun. Instability of incompressible thermo-hyperelastic tubes[J]. Acta Mechanica Sinica, 2007, 39(2):283–288 (in Chinese).Google Scholar
  9. [9]
    Shah A D, Humphrey J D. Finite strain elastodynamics of intracranial aneurysms[J]. J Biomech, 1999, 32(3):593–595.CrossRefGoogle Scholar
  10. [10]
    Guo Z H, Solecki R. Free and forced finite amplitude oscillations of an elastic thick-walled hollow sphere made of incompressible material[J]. Arch Mech Stos, 1963, 15(3):427–433.zbMATHMathSciNetGoogle Scholar
  11. [11]
    Calderer C. The dynamical behavior of nonlinear elastic spherical shells[J]. J Elasticity, 1983, 13(1):17–47.zbMATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    Haslach A D, Humphrey J D. Dynamics of biological soft tissue and rubber: internally pressurized spherical membranes surrounded by a fluid[J]. Int J Non-Linear Mech, 2004, 39(3):399–420.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH 2008

Authors and Affiliations

  1. 1.Department of Mechanics, Shanghai Institute of Applied Mathematics and MechanicsShanghai UniversityShanghaiP. R. China

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